{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Interactive eigenvectors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots, Interact"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "plot_image (generic function with 2 methods)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function plot_image(M, x, tol=0.05)\n",
    "    \n",
    "    new_x = M * x\n",
    "    \n",
    "    unit_circle = [(cos(θ), sin(θ)) for θ in 0:0.01:2π]\n",
    "    ellipse = [M*[cos(θ), sin(θ)] for θ in 0:0.01:2π];\n",
    "\n",
    "    \n",
    "    plot([(0, 0), (x[1], x[2])], aspect_ratio=1, lw=2, leg=false, c=:green, arrow=0.4)\n",
    "    plot!([(0, 0), (new_x[1], new_x[2])], lw=3, c=:blue, arrow=0.4)\n",
    "\n",
    "    plot!(unit_circle, ls=:dash, c=:green)\n",
    "    plot!(first.(ellipse), last.(ellipse), ls=:dash, c=:blue)\n",
    "\n",
    "\n",
    "    # highlight when an eigenvector by changing color:\n",
    "    \n",
    "    ang1 = atan(x[2], x[1])\n",
    "    ang2 = atan(new_x[2], new_x[1])\n",
    "    \n",
    "    if abs(ang1 - ang2) < tol || abs(ang1 - ang2 + π) < tol || abs(ang1 - ang2 - π) < tol\n",
    "        plot!([(0, 0), (new_x[1], new_x[2])], lw=4, c=:magenta)\n",
    "        plot!([(0, 0), (x[1], x[2])], lw=4, c=:magenta)\n",
    "    end\n",
    "    \n",
    "    scatter!([0], [0])\n",
    "\n",
    "    xlims!(-2, 2)\n",
    "    ylims!(-2, 2)\n",
    "end"
   ]
  },
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       "            window,\n",
       "        );\n",
       "    } else {\n",
       "        document\n",
       "            .querySelector('[data-webio-mountpoint=\"13504387864273198495\"]')\n",
       "            .innerHTML = '<strong>WebIO not detected.</strong>';\n",
       "    }\n",
       "    </script>\n",
       "</div>\n"
      ],
      "text/plain": [
       "(div { style=Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"column\") }\n",
       "  (div { style=Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"row\") }\n",
       "    (div { className=\"field interact-widget\" }\n",
       "            (div { className=\"interact-flex-row interact-widget\" }\n",
       "        (div { className=\"interact-flex-row-left\" }\n",
       "          (label { className=\"interact \" style=Dict{Any,Any}(:padding=>\"5px 10px 0px 10px\") }\n",
       "            \"a\"))\n",
       "        (div { className=\"interact-flex-row-center\" }\n",
       "          (input { max=41 min=1 attributes=Dict{Any,Any}(:type=>\"range\",Symbol(\"data-bind\")=>\"numericValue: index, valueUpdate: 'input', event: {change: function (){this.changes(this.changes()+1)}}\",\"orient\"=>\"horizontal\") step=1 className=\"slider slider is-fullwidth\" style=Dict{Any,Any}() }))\n",
       "        (div { className=\"interact-flex-row-right\" }\n",
       "          (p { attributes=Dict(\"data-bind\"=>\"text: formatted_val\") }))))\n",
       "    (div { className=\"field interact-widget\" }\n",
       "            (div { className=\"interact-flex-row interact-widget\" }\n",
       "        (div { className=\"interact-flex-row-left\" }\n",
       "          (label { className=\"interact \" style=Dict{Any,Any}(:padding=>\"5px 10px 0px 10px\") }\n",
       "            \"b\"))\n",
       "        (div { className=\"interact-flex-row-center\" }\n",
       "          (input { max=41 min=1 attributes=Dict{Any,Any}(:type=>\"range\",Symbol(\"data-bind\")=>\"numericValue: index, valueUpdate: 'input', event: {change: function (){this.changes(this.changes()+1)}}\",\"orient\"=>\"horizontal\") step=1 className=\"slider slider is-fullwidth\" style=Dict{Any,Any}() }))\n",
       "        (div { className=\"interact-flex-row-right\" }\n",
       "          (p { attributes=Dict(\"data-bind\"=>\"text: formatted_val\") })))))\n",
       "  (div { style=Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"row\") }\n",
       "    (div { className=\"field interact-widget\" }\n",
       "            (div { className=\"interact-flex-row interact-widget\" }\n",
       "        (div { className=\"interact-flex-row-left\" }\n",
       "          (label { className=\"interact \" style=Dict{Any,Any}(:padding=>\"5px 10px 0px 10px\") }\n",
       "            \"c\"))\n",
       "        (div { className=\"interact-flex-row-center\" }\n",
       "          (input { max=41 min=1 attributes=Dict{Any,Any}(:type=>\"range\",Symbol(\"data-bind\")=>\"numericValue: index, valueUpdate: 'input', event: {change: function (){this.changes(this.changes()+1)}}\",\"orient\"=>\"horizontal\") step=1 className=\"slider slider is-fullwidth\" style=Dict{Any,Any}() }))\n",
       "        (div { className=\"interact-flex-row-right\" }\n",
       "          (p { attributes=Dict(\"data-bind\"=>\"text: formatted_val\") }))))\n",
       "    (div { className=\"field interact-widget\" }\n",
       "            (div { className=\"interact-flex-row interact-widget\" }\n",
       "        (div { className=\"interact-flex-row-left\" }\n",
       "          (label { className=\"interact \" style=Dict{Any,Any}(:padding=>\"5px 10px 0px 10px\") }\n",
       "            \"d\"))\n",
       "        (div { className=\"interact-flex-row-center\" }\n",
       "          (input { max=41 min=1 attributes=Dict{Any,Any}(:type=>\"range\",Symbol(\"data-bind\")=>\"numericValue: index, valueUpdate: 'input', event: {change: function (){this.changes(this.changes()+1)}}\",\"orient\"=>\"horizontal\") step=1 className=\"slider slider is-fullwidth\" style=Dict{Any,Any}() }))\n",
       "        (div { className=\"interact-flex-row-right\" }\n",
       "          (p { attributes=Dict(\"data-bind\"=>\"text: formatted_val\") }))))))"
      ]
     },
     "execution_count": 4,
     "metadata": {
      "application/vnd.webio.node+json": {
       "kernelId": "b60d3081-e7dd-4d68-bb83-50e4fad3e468"
      }
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = slider(-2:0.1:2, label=\"a\")\n",
    "b = slider(-2:0.1:2, label=\"b\")\n",
    "c = slider(-2:0.1:2, label=\"c\")\n",
    "d = slider(-2:0.1:2, label=\"d\")\n",
    "\n",
    "\n",
    "sliders = vbox( hbox(a, b),\n",
    "                hbox(c, d)\n",
    "            )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
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       "            window,\n",
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       "    } else {\n",
       "        document\n",
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       "            .innerHTML = '<strong>WebIO not detected.</strong>';\n",
       "    }\n",
       "    </script>\n",
       "</div>\n"
      ],
      "text/plain": [
       "Observable{Any} with 0 listeners. Value:\n",
       "Plot{Plots.GRBackend() n=7}"
      ]
     },
     "execution_count": 5,
     "metadata": {
      "application/vnd.webio.node+json": {
       "kernelId": "b60d3081-e7dd-4d68-bb83-50e4fad3e468"
      }
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r = -2:0.01:2\n",
    "\n",
    "θ_vals = range(-2π, 2π, length = 500)\n",
    "formatted_vals = map(t -> \"$(floor(t/π, digits=2))π\", θ_vals)\n",
    "\n",
    "#@manipulate for a in r, b in r, c in r, d in r, θ in \n",
    "\n",
    "θ = slider(θ_vals, formatted_vals, label=\"θ\")\n",
    "\n",
    "p = map(a, b, c, d, θ) do a, b, c, d, θ\n",
    "\n",
    "    M = [a b; c d]\n",
    "   \n",
    "    x = [cos(θ), sin(θ)]\n",
    "    \n",
    "    plot_image(M, x)\n",
    "\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
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   "metadata": {},
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   "source": [
    "vbox(sliders, θ, p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Define matrix using the SVD"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "using LinearAlgebra"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
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(WebIO.setval({\\\"name\\\":\\\"changes\\\",\\\"scope\\\":\\\"knockout-component-895d9bd1-ec8f-48b6-9f6f-9d8398dac5e4\\\",\\\"id\\\":\\\"ob_50\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"changes\\\"]=false}),self),this[\\\"index\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"index\\\"]) ? 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   var self = this;\\n    function AppViewModel() {\\n        for (var key in json_data) {\\n            var el = json_data[key];\\n            this[key] = Array.isArray(el) ? ko.observableArray(el) : ko.observable(el);\\n        }\\n        \\n        [this[\\\"formatted_val\\\"]=ko.computed(    function(){\\n        return this.formatted_vals()[parseInt(this.index())-1];\\n    }\\n,this)]\\n        [this[\\\"changes\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"changes\\\"]) ? (WebIO.setval({\\\"name\\\":\\\"changes\\\",\\\"scope\\\":\\\"knockout-component-7f597404-8eca-4321-bfe3-e01dc7aa370a\\\",\\\"id\\\":\\\"ob_53\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"changes\\\"]=false}),self),this[\\\"index\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"index\\\"]) ? 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   var self = this;\\n    function AppViewModel() {\\n        for (var key in json_data) {\\n            var el = json_data[key];\\n            this[key] = Array.isArray(el) ? ko.observableArray(el) : ko.observable(el);\\n        }\\n        \\n        [this[\\\"formatted_val\\\"]=ko.computed(    function(){\\n        return this.formatted_vals()[parseInt(this.index())-1];\\n    }\\n,this)]\\n        [this[\\\"changes\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"changes\\\"]) ? 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(WebIO.setval({\\\"name\\\":\\\"index\\\",\\\"scope\\\":\\\"knockout-component-c37a04b4-0615-4aee-88ba-6db78ee26adb\\\",\\\"id\\\":\\\"ob_55\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"index\\\"]=false}),self)]\\n        \\n    }\\n    self.model = new AppViewModel();\\n    self.valueFromJulia = {};\\n    for (var key in json_data) {\\n        self.valueFromJulia[key] = false;\\n    }\\n    ko.applyBindings(self.model, self.dom);\\n}\\n);\\n    (WebIO.importBlock({\\\"data\\\":[{\\\"name\\\":\\\"knockout\\\",\\\"type\\\":\\\"js\\\",\\\"url\\\":\\\"/assetserver/f07383ad7ba5d9e293900fdc7e5e611110d6ff32-knockout.js\\\"},{\\\"name\\\":\\\"knockout_punches\\\",\\\"type\\\":\\\"js\\\",\\\"url\\\":\\\"/assetserver/306cc87ce82e45c3e5eb81524cb970ae2bab3932-knockout_punches.js\\\"}],\\\"type\\\":\\\"async_block\\\"})).then((imports) => handler.apply(this, imports));\\n}\\n\")])], Dict{Symbol,Any}(:className=>\"field interact-widget\"), 1), Observable{Any} with 0 listeners. Value:\n",
       "Node{DOM}(DOM(:html, :div), Any[Plot{Plots.GRBackend() n=7}], Dict{Symbol,Any}(:className=>\"interact-flex-row interact-widget\"), 1)], Dict{Symbol,Any}(), 6)"
      ]
     },
     "execution_count": 11,
     "metadata": {
      "application/vnd.webio.node+json": {
       "kernelId": "b60d3081-e7dd-4d68-bb83-50e4fad3e468"
      }
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@manipulate for σ₁ in 0:0.01:2, σ₂ in 0:0.01:2, θ in 0:0.01:2π, ϕ in 0:0.01:2π, angle=-2π:0.01:2π\n",
    "\n",
    "    # when phi=theta have U = V so a symmetric matrix\n",
    "    \n",
    "    U = [cos(θ) sin(θ); -sin(θ) cos(θ)]\n",
    "    V = [cos(ϕ) sin(ϕ); -sin(ϕ) cos(ϕ)]\n",
    "\n",
    "    Σ = Diagonal([σ₁, σ₂])\n",
    "\n",
    "    M = U * Σ * V'\n",
    "   \n",
    "    x = [cos(angle), sin(angle)]\n",
    "\n",
    "    plot_image(M, x, 0.01)\n",
    "    \n",
    "    title!(\"evals = $(eigvals(M))\")\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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